Context: The car race has just ended ten minutes ago. He not only won the match but broke his own record. Breaking a record in sport takes only a moment. And it just makes sense to me that we are not allowed to use the past continuous ten minutes after someone has broken a record. Is it correct to say “He was breaking his own record” ten minutes after a victory?
He was breaking his own record is used when you want to tell about a series of events that happened in the past as in a story, like "He was breaking his own record when he broke his leg." Perhaps the 10 minute rule of which you speak applies to something like "He has broken his own record!" That is something a sports announcer would say, and yes, it would be strange to say it after 10 minutes had gone by. Later in the day one would say "He broke his own record."
"He has broken his own record" sounds proper right after he breaks it.
"He is on his way to breaking his own record" or "He is on pace to break his own record" or even "He should break his own record" would be correct when it looks like he will, but he hasn't.
"He was breaking his own record" also implies he was on his way to breaking his own record, but he didn't quite succeed.